ALGEBRA I CURRICULUM

Transcription

1 MIDDLETOWN PUBLIC SCHOOLS ALGEBRA I CURRICULUM Grades 8-10 January 2012

2 2/20/2012 Middletwn Public Schls 1

3 T he Middletwn Public Schls Mathematics Curriculum fr grades K-12 was cmpleted in January 2012 by a K-12 team f teachers. The team, identified as the Mathematics Task Frce and Mathematics Curriculum Writers referenced extensive resurces t design the dcument that included: Cmmn Cre State Standards fr Mathematics Cmmn Cre State Standards fr Mathematics, Appendix A Understanding Cmmn Cre State Standards, Kendall PARCC Mdel Cntent Framewrks Numerus state curriculum Cmmn Cre framewrks, e.g. Ohi Department f Educatin High Schl Traditinal Plus Mdel Curse Sequence, Achieve, Inc. Grade Level and Grade Span Expectatins (GLEs/GSEs) fr Mathematics Third Internatinal Mathematics and Science Test (TIMSS) Best Practice, New Standards fr Teaching and Learning in America s Schls; Differentiated Instructinal Strategies Instructinal Strategies That Wrk, Marzan Gals fr the district Missin Statement Our missin is t prvide a sequential and cmprehensive K-12 mathematics curriculum in a cllabrative student centered learning envirnment that develps critical thinkers, skillful prblem slvers, and effective cmmunicatrs f mathematics. The Middletwn Public Schls Mathematics Curriculum identifies what students shuld knw and be able t d in mathematics. Each grade r curse includes Cmmn Cre State Standards (CCSS), Grade Level Expectatins (GLEs), Grade Span Expectatins (GSEs), grade level supprtive tasks, teacher ntes, best practice instructinal strategies, resurces, a map (r suggested timeline), rubrics, checklists, and cmmn frmative and summative assessments. COMMON CORE STATE STANDARDS The Cmmn Cre State Standards (CCSS): Are fewer, higher, deeper, and clearer. Are aligned with cllege and wrkfrce expectatins. Include rigrus cntent and applicatins f knwledge thrugh high-rder skills. Build upn strengths and lessns f current state standards (GLEs and GSEs). Are internatinally benchmarked, s that all students are prepared fr succeeding in ur glbal ecnmy and sciety. Are research and evidence-based. Cmmn Cre State Standards cmpnents include: Standards fr Mathematical Practice (K-12) Standards fr Mathematical Cntent: Categries (high schl nly): e.g. numbers, algebra, functins, data Dmains: larger grups f related standards Clusters: grups f related standards Standards: define what students shuld understand and are able t d The Middletwn Public Schls Cmmn Cre Mathematics Curriculum prvides all students with a sequential cmprehensive educatin in mathematics thrugh the study f: Standards fr Mathematical Practice (K-12) Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics* Use apprpriate tls Attend t precisin 2/20/2012 Middletwn Public Schls 2

4 Lk fr and make use f Lk fr and express reasning Standards fr Mathematical Cntent: K 5 Grade Level Dmains f Cunting and Cardinality Operatins and Algebraic Thinking Number and Operatins in Base Ten Number and Operatins Fractins Measurement and Data Gemetry 6-8 Grade Level Dmains f Ratis and Prprtinal Relatinships The Number System Expressins and Equatins Functins Gemetry 9-12 Grade Level Cnceptual Categries f Number and Quantity Algebra Functins Mdeling Gemetry Statistics and Prbability RESEARCH-BASED The Middletwn Public Schls Cmmn Cre Mathematics Curriculum prvides a list f research-based best practice instructinal strategies that the teacher may mdel and/r facilitate. It is suggested the teacher: Use frmative assessment t guide instructin Prvide pprtunities fr independent, partner and cllabrative grup wrk Differentiate instructin by varying the cntent, prcess, and prduct and prviding pprtunities fr: anchring cubing jig-sawing pre/pst assessments tiered assignments Address multiple intelligences instructinal strategies, e.g. visual, bdily kinesthetic, interpersnal Prvide pprtunities fr higher level thinking: Webb s Depth f Knwledge, 2,3,4, skill/cnceptual understanding, strategic reasning, extended reasning Facilitate the integratin f Mathematical Practices in all cntent areas f mathematics Facilitate integratin f the Applied Learning Standards (SCANS): cmmunicatin critical thinking prblem slving reflectin/evaluatin research 2/20/2012 Middletwn Public Schls 3

5 Emply strategies f best practice (student-centered, experiential, hlistic, authentic, expressive, reflective, scial, cllabrative, demcratic, cgnitive, develpmental, cnstructivist/heuristic, and challenging) Prvide rubrics and mdels Address multiple intelligences and brain dminance (spatial, bdily kinesthetic, musical, linguistic, intrapersnal, interpersnal, mathematical/lgical, and naturalist) Emply mathematics best practice strategies e.g. using manipulatives facilitating cperative grup wrk discussing mathematics questining and making cnjectures justifying f thinking writing abut mathematics facilitating prblem slving apprach t instructin integrating cntent using calculatrs and cmputers facilitating learning using assessment t mdify instructin COMMON The Middletwn Public Schls Cmmn Cre Mathematics Curriculum includes cmmn assessments. Required (red ink) indicates the assessment is required f all students e.g. cmmn tasks/perfrmance-based tasks, standardized mid-term exam, standardized final exam. Required Assessments Benchmark Prblems Mid-Term Assessment Final Exam Cmmn Prtfli Tasks (2 Anchr Tasks Per Year, HS) NWEA Test Cmmn Instructinal Assessments (I) - used by teachers and students during the instructin f CCSS. Cmmn Frmative Assessments (F) - used t measure hw well students are mastering the cntent standards befre taking state assessments teacher and student use t make decisins abut what actins t take t prmte further learning n-ging, dynamic prcess that invlves far mre frequent testing serves as a practice fr students Cmmn Summative Assessment (S)- used t measure the level f student, schl, r prgram success make sme srt f judgment, e.g. what grade prgram effectiveness e.g. state assessments (AYP), mid-year and final exams Additinal assessments include: Anecdtal recrds Oral presentatins Cnferencing Prblem/Perfrmance based/cmmn tasks Exhibits Rubrics/checklists (mathematical practice, mdeling) Interviews Tests and quizzes Graphic rganizers Technlgy Jurnals Think-aluds Mathematical Practices Writing genres Mdeling Arguments/ pinin Multiple Intelligences assessments, e.g. Infrmative Rle playing - bdily kinesthetic Narrative Graphic rganizing - visual Research Cllabratin - interpersnal 2/20/2012 Middletwn Public Schls 4

6 FOR ALGEBRA I Textbks Algebra I, McDugal Littel (HS) Algebra, Tls fr a Challenging Wrld Prentice Hall (grade 8) Impact Mathematics Algebra and Mre fr the Middle Grades, Everyday Learning Crpratin Supplementary Classrm Instructin That Wrks, Marzan Exemplars (grade 8) NECAP, MCAS, NAEP Released Tasks NWEA MAP Assessments Technlgy Cmputer lab Cmputers Dcument camera ELMO Graphing calculatr Interactive bards LCD prjectrs MIMIO Overhead scientific calculatr Scientific calculatr Smart bard TI Navigatr Websites explrelearning.cm (Gizm ) cm/ Materials Algebra tiles Clred pencils Exp markers Graph paper Overhead Algebra tiles Rulers Student white/graph bards 2/20/2012 Middletwn Public Schls 5

7 NUMBER AND QUANTITY The Real Number System (N-RN) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls Students extend the prperties f expnents t ratinal expnents. N-RN.1 Explain hw the definitin f the meaning f ratinal expnents fllws frm extending the prperties f integer expnents t thse values, allwing fr a ntatin fr radicals in terms f ratinal expnents. Fr example, we define 5 1/3 t be the cube rt f 5 because we want (5 1/3 ) 3 = 5( 1/3 ) 3 t hld, s (5 1/3 ) 3 must equal 5. N-RN.2 Rewrite expressins invlving radicals and ratinal expnents using the prperties f expnents. remved fr public viewing Simplify radicals. Ratinalize denminatr. NUMBER AND QUANTITY The Real Number System (N- RN). Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning NUMBER AND QUANTITY Students use prperties f ratinal and irratinal numbers. N-RN.3 Explain why the sum r prduct f tw ratinal numbers is ratinal that the sum f a ratinal number and an irratinal number is irratinal and that the prduct f a nnzer ratinal number and an irratinal number is irratinal. remved fr public viewing Students reasn and use units t slve prblems. 2/20/2012 Middletwn Public Schls 6

8 Middletwn Public Schls Quantities (N-Q) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning N-Q.1 Use units as a way t understand prblems and t guide the slutin f multistep prblems; chse and interpret units cnsistently in frmulas; chse and interpret the scale and the rigin in graphs and data displays. Use units f measure apprpriately and cnsistently when slving prblems acrss cntent strands and makes cnversins within r acrss systems (NOT TO BE TAUGHT AS A SEPARATE, belw is cnsidered t be prir knwledge) Length Units (accuracy): Inch (t 1/16 inch); Ft; Centimeter (t 1/10 centimeter); Meter (t 1/100 meter); Yard; Mile (use in scale and rate questins); Kilmeter (use in scale and rate questins) Equivalencies: 12 inches in 1 ft; 100 centimeters in 1 meter; 3 feet in 1 yard; 36 inches in 1 yard; 10 millimeters in 1 centimeter Time Unit (accuracy): Hur (t 1 minute); Day; Year Equivalencies: 24 hurs in 1 day; 7 days in 1 week; 365 days in 1 year; 60 secnds in 1 minute; 60 minutes in 1 hur Temperature Unit (accuracy): ºC and º F (t 1 degree) Capacity Unit (accuracy): Quarts (t 1 unce); Galln; Pint; Liter Equivalencies: 32 unces in 1 quart; 4 quarts in 1 galln; 2 pints in 1 quart; 1000 milliliters in 1 liter Unit (accuracy): Kilgram; Gram (t 1/10 gram) Weight Unit (accuracy): Pund (t 1 unce) Equivalencies: 16 unces in 1 pund Angle and Rtatin Unit (accuracy): Degree (t 2 degrees) (2.6.1) N-Q.2 Define apprpriate quantities fr the purpse f descriptive mdeling. N-Q.3 Chse a level f accuracy apprpriate t limitatins n measurement when reprting quantities. remved fr public viewing Fundatin fr wrk with expressins, equatins and functins ALGEBRA Students Interpret the f expressins. 2/20/2012 Middletwn Public Schls 7

9 Seeing in Expressins (A-SSE) Middletwn Public Schls A-SSE.1 Interpret expressins that represent a quantity in terms f its cntext. a. Interpret parts f an expressin, such as terms, factrs, and cefficients. (A-SSE.1a) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning b. Interpret cmplicated expressins by viewing ne r mre f their parts as a single entity. Fr example, interpret P(1+r) n as the prduct f P and a factr nt depending n P. (A-SSE.1b) Identify, extend, and generalize a variety f patterns (linear and nnlinear) represented by mdels, tables, rdered pairs, sequences, graphs t slve prblem. (state assessment) (F&A) 10 1 (3.3.1) A-SSE.2 Use the f an expressin t identify ways t rewrite it. Fr example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recgnizing it as a difference f squares that can be factred as (x 2 y 2 )(x 2 + y 2 ). Linear, expnential, quadratic Demnstrate cnceptual understanding f algebraic expressins by a. adding, subtracting, multiplying and dividing plynmials b. factring quadratic plynmials including difference f squares (grade 8) and HS (D) (F&A-12-3) (3.3.2) remved fr public viewing ALGEBRA Students write expressins in equivalent frms t slve prblems. Seeing in Expressins (A- SSE) A-SSE.3 Chse and prduce an equivalent frm f an expressin t reveal and explain prperties f the quantity represented by the expressin. a. Factr a quadratic expressin t reveal the zers f the functin it Quadratic and expnential defines. (A-SSE.3a) Demnstrate cnceptual understanding f algebraic Practices t expressins by Make sense f prblems and a. adding, subtracting, multiplying and dividing persevere in slving them Reasn abstractly and plynmials b. factring quadratic plynmials including difference Cnstruct viable arguments and critique the reasning f f squares (grade 8) and HS (D) (F&A-12-3) (3.3.2) thers Mdel with mathematics b. Cmplete the square in a quadratic expressin t reveal the Use apprpriate tls maximum r minimum value f the functin it defines. (A-SSE.3b) 2/20/2012 Middletwn Public Schls 8

10 Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls c. Use the prperties f expnents t transfrm expressins fr expnential functins. Fr example the expressin 1.15 t can be rewritten as (1.15 1/12 ) 12t t t reveal the apprximate equivalent mnthly interest rate if the annual rate is 15%. (A-SSE.3c) remved fr public viewing ALGEBRA Students perfrm arithmetic peratins n plynmials. Arithmetic with plynmials and ratinal expressins (A-APR) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning A- APR.1 Understand that plynmials, linear and quadratic frm a system analgus t the integers, namely, they are clsed under the peratins f additin, subtractin, and multiplicatin; add, subtract, and multiply plynmials. Demnstrate cnceptual understanding f algebraic expressins by a. adding, subtracting, multiplying and dividing plynmials b. factring quadratic plynmials including difference f squares (grade 8) and HS (D) (F&A-12-3) (3.3.2) remved fr public viewing Linear and quadratic ALGEBRA Creating Equatins (A- CED) Students create equatins that describe numbers r relatinships. A-CED.1 Create equatins and inequalities in ne variable and use them t slve prblems. Include equatins arising frm linear and quadratic functins and expnential functins. (integers and inputs nly) Demnstrate cnceptual understanding f equality by a. slving prblems invlving algebraic reasning abut equality 2/20/2012 Middletwn Public Schls 9

11 Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls and inequality, e.g. multi-step equatins and systems f equatins (D) b. translating prblem situatins int equatins c. slving linear equatins (symblically and graphically) and expressing the slutin set symblically (algebraically) graphically by prviding the meaning (verbally) f the graphical interpretatins f slutin(s) in prblem-slving situatins d. slving prblems invlving systems f linear equatins in a cntext algebraically r graphically e. slving prblems using mdels r representatins. (state assessment) (F&A) 10 4 (3.4.1) A-CED.2 Create equatins in tw r mre variables t represent relatinships between quantities; graph equatins n crdinate axes with labels and scales. (linear, quadratic and expnential (integer inputs nly) GSE same as abve (F&A) 10 4 (3.4.1) A-CED.3 Represent cnstraints by equatins r inequalities, and by systems f equatins and/r inequalities, and interpret slutins as viable r nnviable ptins in a mdeling cntext. (linear nly) Fr example, represent inequalities describing nutritinal and cst cnstraints n cmbinatins f different fds. GSE same as abve (F&A) 10 4 (3.4.1) A-CED.4 Rearrange frmulas t highlight a quantity f interest, using the same reasning as in slving equatins. (linear, quadratic, and expnential ( i integer inputs nly)) Fr example, rearrange Ohm s law V = IR t highlight resistance R. GSE same as abve (F&A) 10 4 (3.4.1) Linear, quadratic, and expnential (integer inputs nly) fr A.CED.3, linear nly remved fr public viewing ALGEBRA Students understand slving equatins as a prcess f reasning and explain the reasning. 2/20/2012 Middletwn Public Schls 10

12 Middletwn Public Schls Reasning with Equatins and Inequalities (A-REI) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning A-REI.1 Explain each step in slving a simple equatin as fllwing frm the equality f numbers asserted at the previus step, starting frm the assumptin that the riginal equatin has a slutin. Cnstruct a viable argument t justify a slutin methd Demnstrate cnceptual understanding f equality by a. slving prblems invlving algebraic reasning abut equality and inequality, e.g. multi-step equatins and systems f equatins (D) b. translating prblem situatins int equatins c. slving linear equatins (symblically and graphically) and expressing the slutin set symblically (algebraically) graphically by prviding the meaning (verbally) f the graphical interpretatins f slutin(s) in prblem-slving situatins d. slving prblems invlving systems f linear equatins in a cntext algebraically r graphically e. slving prblems using mdels r representatins. (state assessment) (F&A) 10 4 (3.4.1) Master linear, learn as general principle ALGEBRA Students slve equatins and inequalities in ne variable. Reasning with Equatins and Inequalities (A-REI) A-REI.3 Slve linear equatins and inequalities in ne variable, including equatins with cefficients represented by letters. Demnstrate cnceptual understanding f equality by a. slving prblems invlving algebraic reasning abut equality and inequality, e.g. multi-step equatins and systems f equatins (D) b. translating prblem situatins int equatins c. slving linear equatins (symblically and graphically) and expressing Practices t Make sense f prblems and the slutin set persevere in slving them Reasn abstractly and symblically (algebraically) graphically Cnstruct viable arguments by prviding the meaning (verbally) f the graphical and critique the reasning f thers interpretatins f slutin(s) in prblem-slving Mdel with mathematics situatins Use apprpriate tls d. slving prblems invlving systems f linear Attend t precisin equatins in a cntext algebraically r graphically Lk fr and make use f e. slving prblems using mdels r representatins. (state Lk fr and express assessment) (F&A) 10 4 (3.4.1) reasning 2/20/2012 Middletwn Public Schls 11

13 Middletwn Public Schls A-REI.4 Slve quadratic equatins in ne variable. a. Use the methd f cmpleting the square t transfrm any quadratic equatin in x int an equatin f the frm (x p) 2 = q that has the same slutins. Derive the quadratic frmula frm this frm. A-REI.4a X 2 = 49 b. Slve quadratic equatins by inspectin (e.g., fr x 2 = 49), taking square rts, cmpleting the square, the quadratic frmula and factring, as apprpriate t the initial frm f the equatin. Linear inequalities; literal equatins that are linear in the variables being slved fr; quadratics with real slutins Recgnize when the quadratic frmula gives cmplex slutins and write them as a ± bi fr real numbers a and b. A-REI.4b ALGEBRA Students analyze and slve linear equatins and pairs f simultaneus linear equatins. (Grade 8 nly) Reasning with Equatins and Inequalities (A-REI) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning 8.EE-8 Analyze and slve pairs f simultaneus linear equatins. a. Understand that slutins t a system f tw linear equatins in tw variables crrespnd t pints f intersectin f their graphs, because pints f intersectin satisfy bth equatins simultaneusly. 8.EE-8a b. Slve systems f tw linear equatins in tw variables algebraically, and estimate slutins by graphing the equatins. Slve simple cases by inspectin. Fr example, 3x + 2y = 5 and 3x +2y = 6 have n slutin because 3x + 2y cannt simultaneusly be 5 and 6. 8.EE-8b c. Slve real-wrld and mathematical prblems leading t tw linear equatins in tw variables. Fr example, given crdinates fr tw pairs f pints, determine whether the line thrugh the first pair f pints intersects the line thrugh the secnd pair. 8.EE-8c 2/20/2012 Middletwn Public Schls 12

14 ALGEBRA Middletwn Public Schls Students slve systems f equatins. Reasning with Equatins and Inequalities (A-REI) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning A-REI.5 Prve that, given a system f tw equatins in tw variables, replacing ne equatin by the sum f that equatin and a multiple f the ther prduces a system with the same slutins. Demnstrate cnceptual understanding f equality by a. slving prblems invlving algebraic reasning abut equality and inequality, e.g. multi-step equatins and systems f equatins (D) b. translating prblem situatins int equatins c. slving linear equatins (symblically and graphically) and expressing the slutin set symblically (algebraically) graphically by prviding the meaning (verbally) f the graphical interpretatins f slutin(s) in prblem-slving situatins d. slving prblems invlving systems f linear equatins in a cntext algebraically r graphically e. slving prblems using mdels r representatins. (state assessment) (F&A) 10 4 A-REI.6 Slve systems f linear equatins exactly and apprximately (e.g., with graphs), fcusing n pairs f linear equatins in tw variables. GSE same as abve (F&A) 10 4 A-REI.7 Slve a simple system cnsisting f a linear equatin and a quadratic equatin in tw variables algebraically and graphically. Fr example, find the pints f intersectin between the line y = 3x and the circle x 2 +y 2 = 3. GSE same as abve (F&A) 10 4 Linear-linear and linearquadratic ALGEBRA Reasning with Equatins and Inequalities (A-REI) Students represent and slve equatins and inequalities graphically. A-REI.10 Understand that the graph f an equatin in tw variables is the set f all its slutins pltted in the crdinate plane, ften frming a curve 2/20/2012 Middletwn Public Schls 13

15 Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls (which culd be a line). Expand the repertire f prf techniques and use thse techniques in mre sphisticated ways. Use infrmal and frmal reasning and prf t explain and justify cnclusins Frmalize mathematical arguments thrugh the use f deductive reasning Use the principle f mathematical inductin Use reasning and prf thrughut classrm discussins independent f the mathematical tpic being studied Recgnize hw reasning and prf influence the f mathematics (6.2) A-REI.11 Explain why the x-crdinates f the pints where the graphs f the equatins y = f(x) and y = g(x) intersect are the slutins f the equatin f(x) = g(x) Find the slutins apprximately, e.g., using technlgy t graph the functins, make tables f values, r find successive apprximatins. Include cases where f(x) and/r g(x) are linear and expnential, Same as abve 6.2 A-REI.12 Graph the slutins t a linear inequality in tw variables as a half-plane (excluding the bundary in the case f a strict inequality), and graph the slutin set t a system f linear inequalities in tw variables as the intersectin f the crrespnding half-planes Demnstrate cnceptual understanding f equality by a. slving prblems invlving algebraic reasning abut equality and inequality, e.g. multi-step equatins and systems f equatins (D) b. translating prblem situatins int equatins c. slving linear equatins (symblically and graphically) and expressing the slutin set symblically (algebraically) graphically by prviding the meaning (verbally) f the graphical interpretatins f slutin(s) in prblem-slving situatins d. slving prblems invlving systems f linear equatins in a cntext algebraically r graphically e. slving prblems using mdels r representatins. (state assessment) (F&A) 10 4 (3.4.1) 2/20/2012 Middletwn Public Schls 14

16 Middletwn Public Schls FUNCTIONS Students define, evaluate, and cmpare functins. (Grade 8 nly) Interpreting functins (F-IF) 8.F.1 Understand that a functin is a rule that assigns t each input exactly ne utput. Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning The graph f a functin is the set f rdered pairs cnsisting f an input and the crrespnding utput.1 Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) 8.F.2 Cmpare prperties f tw functins each represented in a different way (algebraically, graphically, numerically in tables, r by verbal descriptins). Fr example, given a linear functin represented by a table f values and a linear functin represented by an algebraic expressin, determine which functin has the greater rate f change GSE same as abve (F&A) F.3 Interpret the equatin y = mx + b as defining a linear functin, whse graph is a straight line; give examples f functins that are nt linear. Fr example, the functin A = s2 giving the area f a square as a functin f its side length is nt linear because its graph cntains the pints (1,1),(2,4) and (3,9), which are nt n a 2/20/2012 Middletwn Public Schls 15

17 Middletwn Public Schls straight line. GSE same as abve (F&A) 10 2 FUNCTIONS Interpreting functins (F-IF) Students understand the cncept f a functin and use functin ntatin. F-IF.1 Understand that a functin frm ne set (called the dmain) t anther set (called the range) assigns t each element f the dmain exactly ne element f the range. If f is a functin and x is an element f its dmain, then f(x) dentes the utput f f crrespnding t the input x. The graph f f is the graph f the equatin y = f(x). Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) F-IF.2 Use functin ntatin, evaluate functins fr inputs in their dmains, and interpret statements that use functin ntatin in terms f a cntext. GSE same as abve (F&A) 10 2 (F-IF.3) Recgnize that sequences are functins, smetimes defined recursively, whse dmain is a subset f the integers. 2/20/2012 Middletwn Public Schls 16

18 Middletwn Public Schls Fr example, the Fibnacci sequence is defined recursively by f(0) = f(1) = 1 f(n+1) = f(n) +f(n-1) fr n 1. FUNCTIONS Students use functins t mdel relatinships between quantities. (Grade 8 nly) Interpreting functins (F-IF) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning 8.F.4 Cnstruct a functin t mdel a linear relatinship between tw quantities. Determine the rate f change and initial value f the functin frm a descriptin f a relatinship r frm tw (x, y) values, including reading these frm a table r frm a graph. Interpret the rate f change and initial value f a linear functin in terms f the situatin it mdels, and in terms f its graph r a table f values. Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) 8.F.5 Describe qualitatively the functinal relatinship between tw quantities by analyzing a graph (e.g., where the functin is increasing r decreasing, linear r nnlinear). GSE same as abve (F&A) 10 2 Sketch a graph that exhibits the qualitative features f a functin that has been described verbally. GSE same as abve (F&A) /20/2012 Middletwn Public Schls 17

19 Middletwn Public Schls remved fr public viewing FUNCTIONS Students interpret linear, expnential and quadratic functins that arise in applicatins in terms f the cntext. Interpreting functins (F-IF) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning F.IF.4 Fr a functin (linear, expnential and quadratic) that mdels a relatinship between tw quantities, interpret key features f graphs and tables in terms f the quantities, and sketch graphs shwing key features given a verbal descriptin f the relatinship. Key features include: intercepts intervals where the functin is increasing, decreasing, psitive, r negative relative maximums and minimums symmetries end behavir. Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) F.IF.5 Relate the dmain f a functin (linear, expnential and quadratic) t its graph and, where applicable, t the quantitative relatinship it describes. Fr example, if the functin h(n) gives the number f persn-hurs it takes t assemble n engines in a factry, then the psitive integers wuld be an apprpriate dmain fr the functin. GSE same as abve (F&A) /20/2012 Middletwn Public Schls 18

20 Middletwn Public Schls F.IF.6 Calculate and interpret the average rate f change f a functin (linear, expnential and quadratic) (presented symblically r as a table) ver a specified interval. Estimate the rate f change frm a graph. GSE same as abve (F&A) 10 2 Linear, expnential, and quadratic remved fr public viewing FUNCTIONS Students analyze functins using different representatins. Interpreting functins (F-IF) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning F.IF.7 Graph functins expressed symblically and shw key features f the graph, by hand in simple cases and using technlgy fr mre cmplicated cases. a. Graph linear and quadratic functins and shw intercepts, maxima, and minima. (F.IF.7a) b. Graph square rt, cube rt, and piecewise-defined functins, including step functins and abslute value functins. (F.IF.7b) e. Graph expnential functins shwing intercepts and end behavir shwing perid, midline, and amplitude. (F.IF.7e) Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) Linear, expnential, quadratic, abslute value, step, piece-wise-defined F.IF.8 Write a functin defined by an expressin in different but equivalent frms t reveal and explain different prperties f the functin. 2/20/2012 Middletwn Public Schls 19

21 Middletwn Public Schls a. Use the prcess f factring and cmpleting the square in a quadratic functin t shw zers, extreme values, and symmetry f the graph, and interpret these in terms f a cntext. (F.IF.8a) b. Use the prperties f expnents t interpret expressins fr expnential functins. Fr example, identify percent rate f change in functins such as: y = (1.02) t y = (0.97) t y = (1.01) 12t y = (1.2) t/10 and classify them as representing expnential grwth r decay. (F.IF.8b) Demnstrate understanding f the relative magnitude f real numbers by slving prblems invlving number lines r equality and inequality symbls a. rdering r cmparing ratinal numbers b. cmmn irratinal numbers (e.g., 2, π), c. expnents (ratinal bases with integer expnents) (embedded) d. square rts, cube rts (embedded) e. abslute values,(embedded) f. integers (embedded) g. numbers represented in scientific ntatin. N&O) 10 2 (state assessment) (1.1.1) F.IF.9 Cmpare prperties f tw functins each represented in a different way (algebraically, graphically, numerically in tables, r by verbal descriptins). Fr example, given a graph f ne quadratic functin and an algebraic expressin fr anther, say which has the larger maximum. Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing 2/20/2012 Middletwn Public Schls 20

22 Middletwn Public Schls rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) FUNCTIONS Students build a functin that mdels a relatinship between tw quantities. Building Functins (F- BF) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning F-BF.1 Write a linear, expnential r quadratic functin that describes a relatinship between tw quantities. a. Determine an explicit expressin, a recursive prcess, r steps fr calculatin frm a cntext. (F-BF.1a) Chse apprpriate representatins and mathematical language (e.g., spreadsheets, gemetric mdels, algebraic symbls, tables, graphs, matrices) t present ideas clearly and lgically fr a given situatin See a cmmn in mathematical phenmena that cme frm very different cntexts (e.g., the sum f the first n dd natural numbers, the areas f square gardens, and the distance traveled by a vehicle that starts at rest and accelerates at a cnstant rate can be represented by functins f the frm f(x) = ax 2 ). Find representatins that mdel essential features f a mathematical situatin (e.g., cst f pstage can be mdeled by a step-functin) Use representatins as a primary means fr expressing and understanding mre abstract mathematical cncepts ( ) b. Cmbine standard functin types using arithmetic peratins. Fr example, build a functin that mdels the temperature f a cling bdy by adding a cnstant functin t (F-BF.1b) Explain in ral r written frm hw mathematics cnnects t ther disciplines, t daily life, careers, and sciety (e.g., gemetry in art and literature, data analysis in scial studies, and expnential grwth in finance) Explain multiple appraches that lead t equivalent results when slving prblems. ( ) 2/20/2012 Middletwn Public Schls 21

23 Middletwn Public Schls F-BF.2 Write arithmetic and gemetric sequences bth recursively and with an explicit frmula, use them t mdel situatins, and translate between the tw frms. Identify, extend, and generalize a variety f patterns (linear and nnlinear) represented by mdels, tables, rdered pairs, sequences, graphs t slve prblems. (state assessment) (F&A) 10 1 (3.1.1) Linear, expnential and quadratic FUNCTIONS Students build new functins frm existing functins. Building Functins (F- BF) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning F-BF.3 Identify the effect n the graph f replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) fr specific values f k (bth psitive and negative); find the value f k given the graphs. (linear, expnential, quadratic, and abslute value functin) Experiment with cases and illustrate an explanatin f the effects n the graph using technlgy. Include recgnizing even and dd functins frm their graphs and algebraic expressins fr them. Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) F-BF.4 Find inverse linear functins. a. Slve an equatin f the frm f(x) = c fr a simple functin f that has an inverse and write an expressin fr the inverse. (F-BF.4a) Linear, expnential, 2/20/2012 Middletwn Public Schls 22

24 Middletwn Public Schls quadratic, and abslute value; fr F,BF, 4a, linear nly FUNCTIONS Students cnstruct and cmpare linear, quadratic, and expnential mdels and slve prblems. Linear, Quadratic, and Expnential Mdels (F-LE) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning F.LE.1 Distinguish between situatins that can be mdeled with linear functins and with expnential functins. Chse apprpriate representatins and mathematical language (e.g., spreadsheets, gemetric mdels, algebraic symbls, tables, graphs, matrices) t present ideas clearly and lgically fr a given situatin (6.2.1) a. Prve that linear functins grw by equal differences ver equal intervals, and that expnential functins grw by equal factrs ver equal intervals. (F.LE.1a) b. Recgnize situatins in which ne quantity changes at a cnstant rate per unit interval relative t anther. (F.LE.1b) c. Recgnize situatins in which a quantity grws r decays by a cnstant percent rate per unit interval relative t anther. (F.LE.1c) F.LE.2 Cnstruct linear and expnential functins, including arithmetic and gemetric sequences, given a graph, a descriptin f a relatinship, r tw input-utput pairs (include reading these frm a table). Demnstrate cnceptual understanding f linear and nnlinear (quadratic, expnential (D), abslute value (B), step (B) and square rt (B)) functins and relatins (including characteristics f classes f functins) thrugh an analysis f cnstant, variable, r average rates f change intercepts dmain and range maximum and minimum values increasing and decreasing intervals rates f change (e.g., the height is increasing at a decreasing rate) by describing hw change in the value f ne variable relates t change in the value f a secnd variable by wrking between and amng different representatins f functins and relatins (e.g., graphs, tables, equatins, functin ntatin) (state assessment) (F&A) 10 2 (3.2.1) 2/20/2012 Middletwn Public Schls 23

25 Middletwn Public Schls F.LE.3 Observe using graphs and tables that a quantity increasing expnentially eventually exceeds a quantity increasing linearly, quadratically, r (mre generally) as a plynmial functin. FUNCTIONS Students interpret expressins fr functins in terms f the situatin they mdel. Linear, Quadratic, and Expnential Mdels (F-LE) F-LE.5 Interpret the parameters in a linear r expnential functin in terms f a cntext. Find representatins that mdel essential features f a mathematical situatin (e.g., cst f pstage can be mdeled by a step-functin) (6.2.3) Linear and expnential f frm f(x) = b x + K GEOMETRY Students understand and apply the Pythagrean therem. (Grade 8 nly) Gemetric Measurement and Dimensin (8.G) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express 8.G.6 Explain a prf f the Pythagrean Therem and its cnverse. Expand the repertire f prf techniques and use thse techniques in mre sphisticated ways. Use infrmal and frmal reasning and prf t explain and justify cnclusins Frmalize mathematical arguments thrugh the use f deductive reasning Use the principle f mathematical inductin Use reasning and prf thrughut classrm discussins independent f the mathematical tpic being studied Recgnize hw reasning and prf influence the f mathematics ( ) 8.G.7 Apply the Pythagrean Therem t determine unknwn side lengths in right triangles in real-wrld and mathematical prblems in tw and three dimensins Apply the Pythagrean Therem t find a missing side f a right triangle 2/20/2012 Middletwn Public Schls 24

26 reasning Middletwn Public Schls r in prblem slving situatins (embedded and/r grade 8) (G&M) 10 2 (state assessment) (2.2.1) 8.G.8 Apply the Pythagrean Therem t find the distance between tw pints in a crdinate system. Cnnect t radicals, ratinal expnents, and irratinal numbers STATISTICS AND PROBABILITY Students summarize, represent, and interpret data n a single cunt r measurement variable. Interpreting Categrical and Quantitative Data (S-ID) Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning S-ID.1 Represent data with plts n the real number line (dt plts, histgrams, and bx plts). Interpret a given representatin (e.g., bx-and-whisker plts, scatter plts, bar graphs, line graphs, circle graphs, histgrams, frequency charts) t a. make bservatins b. answer questins c. analyze the data t frmulate r justify cnclusins critique cnclusins make predictins slve prblems within mathematics r acrss disciplines r cntexts (e.g. media, wrkplace, scial and envirnmental situatins). (state assessment) (DSP) 10 1 (4.1.1) S-ID.2 Use statistics apprpriate t the shape f the data distributin t cmpare center (median, mean) and spread (interquartile range, standard deviatin) f tw r mre different data sets. Analyze patterns, trends, r distributins in data t slve prblems in a variety f cntexts a. by determining, using, r analyzing measures f central tendency (mean, median, r mde) dispersin (range r variatin) utliers quartile values estimated line f best fit regressin line 2/20/2012 Middletwn Public Schls 25

27 Middletwn Public Schls crrelatin (strng psitive, strng negative, r n crrelatin) b. invlving cnceptual understanding f the sample frm which the statistics were develped. (state assessment) (DSP) 10 2 (4.2.1) S-ID.3 Interpret differences in shape, center, and spread in the cntext f the data sets, accunting fr pssible effects f extreme data pints (utliers). GSE same as abve (DSP) 10 2 (4.2.1) remved fr public viewing STATISTICS AND PROBABILITY Students investigate patterns f assciatin in bivariate data. (Grade 8 nly) Interpreting Categrical and Quantitative Data (8.SP) Investigate patterns f assciatin in bivariate data Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning 8.SP.1 Cnstruct and interpret scatter plts fr bivariate measurement data t investigate patterns f assciatin between tw quantities. Describe patterns such as clustering, utliers, psitive r negative assciatin, linear assciatin, and nnlinear assciatin. Identify r describe representatins r elements f representatins that best display a given set f data r situatin, using bx-and-whisker plts (bx plts) scatter plts bar graphs line graphs circle graphs histgrams frequency charts (state assessment) (DSP) 10 3 (4.3.1) 8.SP.2 Knw that straight lines are widely used t mdel relatinships between tw quantitative variables. Fr scatter plts that suggest a linear assciatin, infrmally fit a straight line, and infrmally assess the mdel fit by judging the clseness f the data pints t the line. GSE same as abve (DSP) 10 3 (4.3.1) 8.SP.3 Use the equatin f a linear mdel t slve prblems in the cntext f bivariate measurement data, interpreting the slpe and intercept. Fr example, in a linear mdel fr a bilgy 2/20/2012 Middletwn Public Schls 26

28 Middletwn Public Schls experiment, interpret a slpe f 1.5 cm/hr as meaning that an additinal hur f sunlight each day is assciated with an additinal 1.5 cm in mature plant height. Find representatins that mdel essential features f a mathematical situatin (e.g., cst f pstage can be mdeled by a step-functin) (6.2.3) 8.SP.4 Understand that patterns f assciatin can als be seen in bivariate categrical data by displaying frequencies and relative frequencies in a twway table. Cnstruct and interpret a tw-way table summarizing data n tw categrical variables cllected frm the same subjects. Use relative frequencies calculated fr rws r clumns t describe pssible assciatin between the tw variables. Fr example, cllect data frm students in yur class n whether r nt they have a curfew n schl nights and whether r nt they have assigned chres at hme. Is there evidence that thse wh have a curfew als tend t have chres? Chse apprpriate representatins and mathematical language (e.g., spreadsheets, gemetric mdels, algebraic symbls, tables, graphs, matrices) t present ideas clearly and lgically fr a given situatin (6.2.2) STATISTICS AND PROBABILITY Students summarize, represent, and interpret data n tw categrical and quantitative variables. Interpreting Categrical and Quantitative Data (S-ID) S-ID.5 Summarize categrical data fr tw categries in tw-way frequency tables. Interpret relative frequencies in the cntext f the data (including jint, marginal, and cnditinal relative frequencies). Practices t Make sense f prblems and persevere in slving them Reasn abstractly and Recgnize pssible assciatins and trends in the data. Identify r describe representatins r elements f representatins that best display a given set f data r situatin, using a. bx-and-whisker plts (bx plts) 2/20/2012 Middletwn Public Schls 27

29 Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics Use apprpriate tls Attend t precisin Lk fr and make use f Lk fr and express reasning Middletwn Public Schls b. scatter plts c. bar graphs d. line graphs e. circle graphs f. histgrams g. frequency charts (state assessment) (DSP) 10 3 (4.3.1) S-ID.6 Represent data n tw quantitative variables n a scatter plt, and describe hw the variables are related. Linear fcus; discuss general principle a. Fit a functin t the data; use functins fitted t data t slve prblems in the cntext f the data. Use given functins r chse a functin suggested by the a cntext. Emphasize linear, quadratic, and expnential mdels. (S-ID.6a) (linear fcus) Chse apprpriate representatins and mathematical language (e.g., spreadsheets, gemetric mdels, algebraic symbls, tables, graphs, matrices) t present ideas clearly and lgically fr a given situatin (6.2.1) b. Infrmally assess the fit f a functin by pltting and analyzing residuals. (S-ID.6b) Identify r describe representatins r elements f representatins that best display a given set f data r situatin, using a. bx-and-whisker plts (bx plts) b. scatter plts c. bar graphs d. line graphs e. circle graphs f. histgrams g. frequency charts (state assessment) (DSP) 10 3 c. Fit a linear functin fr a scatter plt that suggests a linear assciatin. (S-ID.6c) Expand the repertire f prblem-slving strategies and use thse strategies in mre sphisticated ways a. identify relevant infrmatin, questin/task b. chse ne r mre strategies/ representatins make an rganized list create a diagram/picture/chart 2/20/2012 Middletwn Public Schls 28